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Dividend Policy, Growth, and the Valuation of Shares Author(s): Merton H. Miller and Franco Modigliani Source: The Journal of Business, Vol. 34, No. 4 (Oct., 1961), pp. 411-433 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/2351143 Accessed: 06-06-2015 17:38 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Business. http://www.jstor.org This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:38:51 UTC All use subject to JSTOR Terms and Conditions THE JOURNAL OF BUSINESS The GraduateSchool of Business of the University of Chicago VOL.XXXIV OCTOBER1961 No. 4 DIVIDEND POLICY, GROWTH, AND THE VALUATION OF SHARES* MERTON H. MILLERt AND FRANCO MODIGLINIt Tz i~xeffect of a firm'sdividendpolicy on the currentpriceof its sharesis a matter of considerableimportance, not only to the corporate officials who must set the policy, but to investors planning portfolios and to economists seeking to understandand appraise the functioning of the capital markets. Do companies with generous distribution policies consistently sell at a premium over those with -niggardlypayouts? Is the reverseever true? If so, under what conditions? Is there an optimum payout ratio or range of ratios that maximizes the currentworth of the shares? Although these questions of fact have been the subject of many empiricalstudies in recent years no consensushas yet been achieved. One reasonappearsto be the absence in the literature of a complete and reasonablyrigorousstatement of those parts of the economic theory of valuation bearingdirectly on the matter * The authors wish to express their thanks to all who read and commented on earlier versions of this paper and especially to Charles C. Holt, now of the University of Wisconsin, whose suggestions led to considerable simplification of a number of the proofs. t Professor of finance and economics, University of Chicago. t Professor of economics, Northwestern University. of dividendpolicy. Lackingsuch a statement, investigators have not yet been able to frame their tests with sufficient precision to distinguish adequately between the various contending hypotheses. Nor have they been able to give a convincingexplanationof what their test results do imply about the underlying process of valuation. In the hope that it may help to overcome these obstacles to effective empirical testing, this paper will attempt to fill the existing gap in the theoreticalliterature on valuation.We shall begin, in Section I, by examiningthe effects of differences in dividend policy on the current price of sharesin an ideal economy characterizedby perfect capital markets, rational behavior, and perfect certainty. Still within this convenient analytical frameworkwe shall go on in Sections II and III to considercertaincloselyrelated issues that appear to have been responsible for considerablemisunderstanding of the role of dividendpolicy. In particular, Section II will focus on the longstanding debate about what investors "really"capitalizewhen they buy shares; and SectionIII on the muchmooted relations betweenprice, the rate of growth of 411 This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:38:51 UTC All use subject to JSTOR Terms and Conditions 412 THE JOURNALOF BUSINESS profits, and the rate of growth of dividends per share. Once these fundamentals have been established,we shall proceed in Section IV to drop the assumption of certainty and to see the extent to which the earlierconclusionsabout dividend policy must be modified.Finally, in Section V, we shall briefly examine the implications for the dividend policy problem of certain kinds of market imperfections. I. EFFECT OF DIVIDEND POLICY WITH PERFECT MARKETS, RATIONAL BEHAVIOR, AND PERFECT CERTAINTY The meaningof the basic assumptions. -Although the terms "perfectmarkets," "rational behavior," and "perfect certainty" are widely used throughouteconomic theory, it may be helpful to start by spelling out the precise meaning of these assumptionsin the presentcontext. 1. In "perfect capital markets," no buyer or seller (or issuer) of securitiesis large enoughfor his transactionsto have an appreciableimpact on the then ruling price. All tradershave equal and costless access to information about the ruling priceand about all otherrelevant characteristics of shares (to be detailed specificallylater). No brokeragefees, transfer taxes, or other transaction costs are incurred when securities are bought, sold, or issued, and there are no tax differentialseither between distributedand undistributed profits or between dividends and capital gains. 2. "Rational behavior" means that investors always prefer more wealth to less and are indifferentas to whether a given incrementto their wealth takes the form of cash payments or an increase in the market value of their holdings of shares. 3. "Perfect certainty" implies complete assuranceon the part of every in- vestor as to the future investment program and the future profits of every corporation. Because of this assurance, there is, among other things, no need to distinguishbetween stocks and bonds as sourcesof funds at this stage of the analysis. We can, therefore, proceed as if there were only a single type of financial instrument which, for convenience, we shall refer to as shares of stock. The fundamental principle of valuation.-Under'these assumptionsthe valuation of all shares would be governedby the followingfundamentalprinciple:the price of each sharemust be such that the rate of return (dividends plus capital gains per dollarinvested) on every share will be the same throughout the market over any given interval of time. That is, if we let dj(t) = dividendsper sharepaid by firmj duringperiodt pj(t) = the price (ex any dividend in t - 1) of a sharein firmj at the start of period t, we must have dj(t) +pj(t+ 1) -pj(t) pj(t) ~~~(1) = p ( t ) independentof j; or, equivalently, pj( t)= [dj(t)+pj(t+)] (2) for eachj and for all t. Otherwise,holders of low-return (high-priced)shares could increase their terminal wealth by selling these shares and investing the proceeds in shares offeringa higher rate of return. This process would tend to drive down the prices of the low-return shares and drive up the prices of high-returnshares until the differential in rates of return had been eliminated. The effectof dividendpolicy.-The im- This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:38:51 UTC All use subject to JSTOR Terms and Conditions THE VALUATIONOF SHARES plications of this principlefor our problem of dividend policy can be seen somewhat more easily if equation (2) is restated in terms of the value of the enterprise as a whole rather than in terms of the value of an individual share. Dropping the firm subscriptj since this will lead to no ambiguity in the present context and letting n(t) = the number of shares of record at thestart of t m(t + 1) = the number of new shares (if any) sold during t at the ex dividendclosingpricep(t + 1), so that n(t + 1) = n(t) + m(t + 1) V(t) = n(t) p(t) = the total value of the enterpriseand D(t) = n(t) d(t) = the total dividends paid duringt to holdersof rec- ordat the start of t, we can rewrite (2) V(t l +,) 1[D(t)+n(t)p(t+1) I 1+0 -1+ (t) [ D(t) + V(t+ 1) -m (t+ 1) p (t+ 1)I. (3) The advantage of restating the fundamental rule in this form is that it brings into sharper focus the three possible routes by which currentdividendsmight affect the current market value of the firm V(t), or equivalently the price of its individual shares, p(t). Current dividends will clearly affect V(t) via the first term in the bracket, D(t). In principle, current dividends might also affect V(t) indirectly via the second term, V(t + 1), the new ex dividend market value. Since V(t + 1) must depend only on future and not on past events, such could be the case, however, only if both (a) V(t + 1) were a function of future dividendpolicy and (b) the current distribution D(t) servedto convey some otherwiseunavail- 413 able informationas to what that future dividendpolicy wouldbe. The first possibility being the relevant one from the standpointof assessingthe effects of dividend policy, it will clarify matters to assume, provisionally,that the future dividend policy of the firm is known and given for t + 1 and all subsequent periods and is independentof the actual dividend decision in t. Then V(t + 1) will also be independent of the current dividend decision, though it may very well be affected by D(t + 1) and all subsequent distributions.Finally, currentdividends can influence V(t) through the third term, -m(t + 1) p(t + 1), the value of new sharessold to outsidersduring the period. For the higher the dividend payout in any period the more the new capital that must be raisedfrom external sources to maintain any desired level of investment. The fact that the dividend decision effects price not in one but in these two conflicting ways-directly via D(t) and inversely via -m(t) p(t + 1)-is, of course,preciselywhy one speaks of there being a dividend policy problem.If the firm raises its dividend in t, given its investment decision,will the increasein the cash payments to the currentholdersbe more or less than enough to offset their lower shareof the terminalvalue? Which is the better strategy for the firm in financingthe investment: to reducedividends and rely on retainedearningsor to raise dividends but float more new shares? In our ideal world at least these and related questions can be simply and immediately answered: the two dividend effects must always exactly cancel out so that the payout policy to be followedin t will have no effect on the price at t. We need only expressm(t+l)1 p(t+1) in terms of D(t) to show that such must This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:38:51 UTC All use subject to JSTOR Terms and Conditions 414 THE JOURNALOF BUSINESS indeed be the case. Specifically, if I(t) is the given level of the firm's investment or increasein its holdingof physical assets in t and if X(t) is the firm's total net profit for the period, we know that the amount of outside capital required will be = I(t) m(t+1)p(t+1) (4) - [X (t) -D (t) ]. Substituting expression (4) into (3), the D(t) cancel and we obtain for the value of the firm as of the start of t V (t)-n (t) p (t) = +p(t)[X 1 (t)-I(t) + V(t+ 1) (5) Since D(t) does not appear directly among the arguments and since X(t), I(t), V(t + 1) and p(t) are all independent of D(t) (either by their nature or by assumption) it follows that the current value of the firmmust be independentof the current dividend decision. Having established that V(t) is unaffected by the current dividend decision it is easy to go on to show that V(t) must also be unaffectedby any futuredividend decisions as well. Such future decisions can influenceV(t) only via their effect on V (t + 1). But we can repeat the reasoning above and show that V(t + 1)-and hence V(t)-is unaffected by dividend policy in t + 1; that V(t + 2)-and hence V(t + 1) and V(t)-is unaffected by dividend policy in t + 2; and so on for as far into the future as we care to look. Thus, we may concludethat given a firm's investment policy, the dividend payout policy it chooses to follow will affect neither the currentprice of its shares nor the total return to its shareholders. Like many other propositionsin economics, the irrelevanceof dividend policy, given investment policy, is "obvious, once you think of it." It is, after all, merely one more instance of the general principle that there are no "financialillusions"in a rationaland perfecteconomic environment. Values there are determined solely by "real" considerationsin this case the earning power of the firm'sassets and its investment policyand not by how the fruits of the earning power are "packaged" for distribution. Obvious as the proposition may be, however,one finds few referencesto it in the extensive literature on the problem.' It is true that the literatureaboundswith statements that in some "theoretical" sense, dividend policy ought not to count; but either that sense is not clearly specified or, more frequently and especially among economists, it is (wrongly) identified with a situation in which the firm's internal rate of return is the same as the external or market rate of return.2 A major source of these and related misunderstandingsof the role of the dividend policy has been the fruitlessconcern and controversy over what investors "really"capitalizewhen they buy shares. We say fruitless because as we shall now proceedto show, it is actually possibleto derive from the basic principleof valuation (1) not merely one, but severalvaluation formulaseach starting from one of the "classical" views of what is being capitalized by investors. Though differing somewhat in outward appearance, the various formulascan be shown to be equivalent in all- essential respects including, of course, their implicationthat dividend policy is irrelevant. While the 1 Apart from the referencesto it in our earlier papers, especially [16], the closest approximation seemsto be that in Bodenborn[1, p. 4921,but even his treatmentof the role of dividendpolicy is not completelyexplicit. (The numbersin bracketsrefer to referenceslisted below,pp. 432-33). 2 See belowp. 424. This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:38:51 UTC All use subject to JSTOR Terms and Conditions THE VALUATIONOF SHARES 415 controveryitself thus turns out to be an as T approachesinfinity4so that (7) can empty one, the different expressionsdo be expressedas have some intrinsic interest since, by T-1 highlighting different combinations of v (O) = rnim (8) variablesthey provideadditionalinsights into the process of valuation and they X [X(t)-I(t)], open alternative lines of attack on some which we shall further abbreviateto of the problemsof empiricaltesting. c II. WHAT DOES THE MARKET "REALLY" CAPITALIZE? In the literatureon valuation one can find at least the following four more or less distinct approachesto the valuation of shares: (1) the discounted cash flow approach; (2) the current earnings plus future investment opportunities approach; (3) the stream of dividends approach; and (4) the stream of earnings approach.To demonstratethat these approachesare, in fact, equivalentit will be helpful to begin by first going back to equation (5) and developing from it a valuation formula to serve as a point of referenceand comparison.Specifically,if we assume, for simplicity, that the market rate of yield p (t) = p for all t,3 then, setting t = 0, we can rewrite (5) as V (O) 1 IX (O)-I (0) ] + 1 +p ( V (1). +-- (6) Since (5) holds for all t, setting t = 1 permits us to express V(1) in terms of V(2) which in turn can be expressedin terms of V(3) and so on up to any arbitrary terminal period T. Carrying out these substitutions, we obtain T-1 V(O) = E(l+p)t+l[X(t)I(t)] +(1+p) V(T). In general,the remainderterm (1 + P)-T. V(T) can be expected to approachzero =2 V(O) t- 1 (1-+I1 (I+ P)t [X(t)-I(t)]. (9) The discounted cash flow approach.- Consider now the so-called discounted cash flow approach familiar in discussions of capital budgeting.There, in valuing any specificmachinewe discount at the market rate of interest the stream of cash receipts generated by the machine; plus any scrap or terminal value of the machine; and minus the stream of cash outlays for direct labor, materials, repairs, and capital additions. The same approach,of course, can also be applied to the firm as a whole which may be thought of in this context as simply a large, composite machine.5 This ap3More generalformulasin which p(t) is allowed to vary with time can alwaysbe derivedfromthose presentedhere merelyby substitutingthe cumbersomeproduct 1L [l+p(r)] for (1+p)t+' TO0 4 The assumptionthat the remainder vanishesis introducedfor the sake of simplicityof exposition only and is in no way essential to the argument. What is essential,of course,is that V(O),i.e., the sum of the two termsin (7), be finite, but this can always be safely assumedin economicanalysis.See below,n. 14. 5 This is, in fact, the approachto valuationnormally takenin economictheorywhen discussingthe value of the assetsof an enterprise,but much more rarely applied, unfortunately,to the value of the liability side. One of the few to apply the approach to the sharesas well as the assetsis Bodenhornin [1], who uses it to derivea formulacloselysimilarto (9) above. This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:38:51 UTC All use subject to JSTOR Terms and Conditions 416 THE JOURNALOF BUSINESS proach amounts to defining the value of investments in real assets that will yield the firm as more than the "normal"(market)rate of return. The latter opportunities, freT-1 quently termed the "good will" of the V(O) = E P) (0 t=O (10) business,may arise,in practice,from any of a number of circumstances (ranging X [E (t-co() Tv (T), +(+p all the way fromspeciallocationaladvantages to patents or other monopolistic where IR(t)representsthe stream of cash advantages). receipts and ()(t) of cash outlays, or, To see how these opportunitiesaffect abbreviating,as above, to the value of the business assume that in co some future period I the firminvests 1(t) v ( = ?) E,_O (1+p),+'(11 1+p teRW [st-(t I (11) dollars. Suppose, further, for simplicity, that starting in the period immediately But we also know, by definition, that following the investment of the funds, [X(t) -I(t)] = [IR(t) -()(t)] since, X(t) the projectsproducenet profitsat a condiffers from IR(t) and 1(t) differs from stant rate of p*(t) per cent of I (t) in each CO(t)merely by the "cost of goods sold" period thereafter.6 Then the present (and also by the depreciationexpense if worth as of t of the (perpetual)stream of we wish to interpretX(t) and I(t) as net profitsgeneratedwill be I(t) p*(t)/p, and rather than gross profits and invest- the "good will" of the projects (i.e., the ment). Hence (11) is formallyequivalent differencebetween worth and cost) will to (9), and the discounted cash flow ap- be proach is thus seen to be an implication I(t)fP-22)-I(t) =1(t) [P P*(t) P P* of the valuation principle for perfect markets given by equation (1). The present worth as of now of this fuThe investmentopportunitiesapproach. ture "good will" is -Consider next the approachto valuaIt P* ( ) p] (1 + p)-+ tion which would seem most natural from the standpoint of an investor proposing to buy out and operate some al- and the present value of all such future ready-going concern. In estimating how opportunitiesis simply the sum much it would be worthwhileto pay for the privilege of operating the firm, the to P amount of dividendsto be paid is clearly not relevant, since the new owner can, Adding in the present value of the (uniwithin wide limits, make the future divi- formperpetual)earnings,X(O),on the asdend stream whatever he pleases. For 8The assumptionthat I(t) yields a uniformperhim the worth of the enterprise,as such, petuity is not restrictivein the present certainty will depend only on: (a) the "normal" context since it is always possible by means of rate of return he can earn by investing simple,present-valuecalculationsto findan equivauniformperpetuityfor any project, whatever his capital in securities (i.e., the market lent the time shape of its actual returns.Note also that rate of return); (b) the earning power of p*(t) is the averagerate of return.If the managersof the physical assets currentlyheld by the the firmarebehavingrationally,they will, of course, p as their cut-off criterion(cf. below p. 418). firm; and (c) the opportunities, if any, use In this event we would have p*(t) > p. The forthat the firmoffersfor makingadditional mulasremainvalid, however,even wherep*(t) p. For if p*(t) = p, then howI t1-1 ( ever large the growth in assets may be, the second term in (12) will be zero and t =1 t = T=O the firm'sprice-earningsratio would not X ( p)-t) + rise above a humdrum i/p. The essence 0 of "growth,"in short, is not expansion, p) -t =X(O) (I1+ f, but the existence of opportunitiesto int =1 vest significant quantities of funds at higher than "normal"rates of return. t=1,2 . ...o CO ___ + Y. t =1 T=O X (+ T-It1 *T) P) +5 {12 )(t 7A valuationformulaanalogousto (12) though derivedand interpretedin a slightly differentway is foundin Bodenhorn[1].Variantsof (12) forcertain specialcases are discussedin Walter[201. This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:38:51 UTC All use subject to JSTOR Terms and Conditions 418 THE JOURNALOF BUSINESS Notice also that if p*(t) (17) X V(t+ 1)] where Dt(t + r) denotes that portion of the total dividendsD(t + r) paid during periodt + r, that accruesto the sharesof =+[D(t) + V(t+ 1) record as of the start of period t (indicated by the subscript). That equation -m(t+ 1) p(t+ 1)], (14) is equivalent to (9) and hence also to (12) is immediately apparent for the which is (3) and which has already been specialcase in whichno outside financing shown to imply both (9) and (12).12 is undertakenafter period t, for in that There are, of course, other ways in case which the equivalence of the dividend approachto the other approachesmight -I(t+=r). -X(t+r) To allow for outside financing,note that we can rewrite (14) as V(t) D(t) 1 +P [ + E dividends rule out X(t) > I(t) but not X(t) 0 per cent of its earnings,howeversmall the value of e. This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:38:51 UTC All use subject to JSTOR Terms and Conditions 420 THE JOURNAL OF BUSINESS have been established, but the method presented has the advantage perhaps of providing some further insight into the reason for the irrelevance of dividend policy. An increasein currentdividends, given the firm'sinvestment policy, must necessarily reduce the terminal value of existing sharesbecausepart of the future dividend stream that would otherwise have accruedto the existing sharesmust be diverted to attract the outside capital from which, in effect, the higher current dividends are paid. Under our basic assumptions,however,p must be the same for all investors,new as well as old. Consequently the market value of the dividends diverted to the outsiders,which is both the value of their contributionand the reductionin terminalvalue of the existing shares, must always be precisely the same as the increase in current dividends. The stream of earnings approach.Contraryto widely held views, it is also possible to develop a meaningful and consistentapproachto valuationrunning in terms of the stream of earningsgenerated by the corporationrather than of the dividend distributionsactually made to the shareholders.Unfortunately, it is also extremely easy to mistate or misinterpretthe earningsapproachas would be the case if the value of the firm were to be defined as simply the discounted sum of future total earnings.'3 The troublewith such a definitionis not, as is often suggested, that it overlooks the fact that the corporationis a separateentity and that these profits cannot freely be withdrawn by the shareholders;but ratherthat it neglects the fact that additional capital must be acquiredat some cost to maintain the future earnings stream at its specifiedlevel. The capital to be raised in any future period is, of course, I(t) and its opportunity cost, no matter how financed, is p per cent per period thereafter. Hence, the current value of the firm under the earnings approach must be stated as In fairness,we shouldpoint out that thereis no one, to our knowledge,who has seriouslyadvanced this view. It is a view whosemainfunctionseemsto be to serve as a "strawman" to be demolishedby thosesupportingthe dividendview. See,e.g., Gordon (9, esp. pp. 102-31.Other writers take as the supposed earnings counter-viewto the dividend approachnot a relationrunningin termsof the stream of earningsbut simply the propositionthat price is proportional to current earnings, i.e., V(O)= X(O)/p. The probable origins of this widespread misconceptionabout the earningsapproachare discussedfurtherbelow (p. 424). 00 co V (0) = f +w+ (18) X [X(t) pI(r)]. - That this version of the earnings approachis indeedconsistentwith ourbasic assumptions and equivalent to the previous approachescan be seen by regrouping terms and rewritingequation (18) as V(0) X(t) So (lp+ 00 00 t=oVS pI (t) (I +p)7+12 00 y 13 ( + p ) t+1 PI (t) Since the last inclosed summation reduces simply to I(t), the expression(19) in turn reduces to simply 0c V(0) = E (- 1 t_+1[X(t)-I This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:38:51 UTC All use subject to JSTOR Terms and Conditions (t)], (20) THE VALUATIONOF SHARES 421 which is precisely our earlier equation and kp* is the (constant) rate of growth of total earnings. Substituting from (21) (9). Note that the version of the earnings into (12) for 1(t) we obtain approachpresentedhere does not depend for its validity upon any special assump- V(O) +_E tions about the time shape of the stream of total profitsor the stream of dividends X kX(O) [ 1 + kp*] t per share. Clearly, however, the time X ( 1 + p)-(t+I) (2 2) paths of the two streams are closely related to each other (via financialpolicy) _x(o) r k (p* -P) and to the stream of returns derived by L 1 + holders of the shares. Since these relaco 1 +k P*>t tions are of some interest in their own X Sk -J right and since misunderstandingsabout them have contributedto the confusion Evaluating the infinite sum and simpliover the role of dividend policy, it may fying, we finally obtain14 be worthwhile to examine them briefly before moving on to relax the basic as- V(O) =-(?) [1 + k(p* p)] p -k p sumptions. p _ III. EARNINGS, DIVIDENDS, AND GROWTH RATES The convenientcase of constantgrowth rates.-The relation between the stream of earningsof the firm and the stream of dividends and of returns to the stockholders can be brought out most clearly by specializing (12) to the case in which investment opportunitiesare such as to generate a constant rate of growth of profits in perpetuity. Admittedly, this case has little empiricalsignificance,but it is convenient for illustrative purposes and has received much attention in the literature. Specifically, suppose that in each period t the firm has the opportunityto invest in real assets a sum 1(t) that is k per cent as large as its total earningsfor the period; and that this investment produces a perpetual yield of p* beginning with the next period.Then, by definition X(t) = X(t- 1) + p*I(t- 1) =X(t-) [I+kp*] -X(O) [I + kp*] (21) X(O) (1 -k) (23) whichexpressesthe value of the firm as a function of its currentearnings, the rate of growthof earnings,the internalrate of return, and the market rate of return.15 14One advantageof the specialization(23) is that it makesit easy to see what is reallyinvolvedin the assumptionhere and throughoutthe paperthat the V(O)given by any of our summationformulasis necessarilyfinite (cf. above, n. 4). In terms of (23) the conditionis clearlykp* p, so that the so-calledgrowthparadoxdisappearsaltogether.If, as we should generallyexpect, (1 + kp*)/(l + p) is close to one, and if T is not too large, the right hand side of (22a) admits of a very convenientapproximation.In this case in fact we can write 1I+P _I +T(kp* - [1 + X(O) p k ( p* P) P-kP* XT(P -kp*) =X( )+ kX (O) p that D(t) = X(O)[1 -kr]); k - kr = the amountof externalcapi- tal raised per period, expressed as a fraction of profitsin the period. Then the present value of the stream of dividends to the original owners will be p) the approximationholding,if, as we shouldexpect, (1 + kp*) and (I + p) are both close to unity. Substitutingthis approximationinto (22a) and simplifying,finallyyields V (0 the price per share? Clearly, the answer will vary depending on whether or not the firm is paying out a high percentage of its earnings and thus relying heavily on outside financing. We can show the nature of this dependence explicitly by making use of the fact that whatever the rate of growth of dividendsper share the present value of the firm by the dividend approach must be the same as by the earningsapproach.Thus let g = the rate of growthof dividends per share, or, what amountsto the same thing, the rate of growthof dividendsaccruingto the shares of the currentholders(i.e., + g]t); Do(t) = Do(O)[1 kr= the fractionof total profits retainedin each period (so ( (22b (1O+g)t Do EO p) D (O) p g (24) X(0O)[ 1-kr] P-g By virtue of the dividend approach we know that (24) must be equal to V(O). If, therefore,we equate it to the righthand side of (23), we obtain X (0)[1 1-kr] X ( O)[ 1-( kr+ ke)] P-g P-~kP* from which it follows that the rate of growth of dividends per share and the The commonsense of (22b)is easy to see. The current valueof a firmis givenby the valueof the earn- rate of growth of the price of a share ing powerof the currentlyheld assetsplus the mar- must bel6 ket value of the specialearningopportunitymulti16 That g is the rate of priceincreaseper shareas plied by the numberof yearsfor whichit is expected well as the rate of growthof dividendsper sharefolto last. This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:38:51 UTC All use subject to JSTOR Terms and Conditions THE VALUATIONOF SHARES g=kp* 1_kr_ kep 1k . (25) Notice that in the extreme case in which all financing is internal (ke = 0 and k = kr), the second term drops out and the first becomes simply kp*. Hence the growth rate of dividends in that special 423 tive kp*, if p* p) then the stream of dividends and price per share must grow over time even though kr = In X(O)[II FiG. 1.-Growth of dividendsper sharein relationto growthin total earnings: A. Total earnings:ln X(t) = ln X(O) + kp*t; B. Total earningsminus capital invested: ln [X(t) - I(t)] = In X(O) [1 - k] + kp*t; Dividendsper share (all financinginternal):ln Do(t) = In D(O)+ gt = In X(O) [1 - k] + kp*t; C. Dividendsper share (somefinancingexternal):ln Do(t) = In D(O)+ gt; D. Dividendsper share (all financingexternal):In Do(t) = In X(O)+ [(k/i - k) (p* - p)]t. case is exactly the same as that of total profits and total value and is proportional to the rate of retention kr. In all other cases, g is necessarilyless than kp* and may even be negative, despite a posi- 0, that is, even though it pays out all its earningsin dividends. The relation between the growth rate of the firm and the growth rate of dividends under various dividend policies is illustrated graphically in Figure 1 in lows from the fact that by (13) and the definition which for maximum clarity the natural of g logarithm of profits and dividends have been plotted against time.'7 E (1 + p)T+ T Line A shows the total earningsof the firm growing through time at the constant rate kp*, the slope of A. Line B P ) + shows the growth of (1) the stream of ( T=-O total earningsminus capital outlays and d(r) =p(O) [1 + t 17 That is, we replaceeach discretecompounding expressionsuch as X(t) = X(O) [1 + kp*]t with its counterpartunder continuousdiscountingX(t) = X(O)ekP*t which, of course, yields the convenient linear relationIn X(t) = In X(O)+ kp*t. This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:38:51 UTC All use subject to JSTOR Terms and Conditions 424 THE JOURNALOF BUSINESS (2) the streamof dividendsto the original owners (or dividends per share) in the special case in which all financingis internal. The slope of B is, of course, the same as that of A and the (constant) differencebetween the curves is simply ln(l - k), the ratio of dividends to profits. Line C shows the growth of dividends per share when the firm uses both internal and external financing.As compared with the pure retention case, the line starts higher but grows more slowly at the rate g given by (25). The higher the payout policy, the higherthe starting position and the slowerthe growth up to the other limiting case of complete external financing,Line D, which starts at ln X(O)and growsat a rate of (k/I - k) . optimum dividend policy for the firm that depends on the internal rate of return. Such a conclusion is almost inevitable if one worksexclusivelywith the assumption, explicit or implicit, that funds for investment come only from retained earnings.For in that case dividend policy is indistinguishable from investment policy; and there is an optimal investment policy which does in general depend on the rate of return. Notice also from (23) that if p* = p and k = kr, the term [1 - kr] can be canceled from both the numerator and the denominator.The value of the firm becomes simply X(O)/p, the capitalized value of current earnings. Lacking a standardmodel for valuation more general than the retainedearningscase it has (P* -P). The special case of exclusivelyinternal been all too easy for many to conclude financing.-As noted above the growth that this droppingout of the payout ratio rate of dividends per share is not the [1 - kr] when p* = p must be what is same as the growth rate of the firm ex- meant by the irrelevance of dividend cept in the special case in which all policy and that V(O) = X(O)/p must financingis internal. This is merely one constitute the "earnings"approach. Still another example of the pitfalls in of a numberof peculiaritiesof this special case on which, unfortunately, many basing argumentson this special case is writers have based their entire analysis. provided by the recent and extensive The reason for the preoccupation with work on valuation by M. Gordon.'8Gorthis special case is far from clear to us. don argues, in essense, that because of Certainlyno one would suggest that it is increasinguncertainty the discount rate the only empiricallyrelevant case. Even p$(t) applied by an investor to a future if the case were in fact the most common, dividend payment will rise with t, where the theorist would still be under an obli- t denotes not a specific date but rather gation to consider alternative assump- the distance from the period in which tions. We suspect that in the last analy- the investor performs the discounting.'9 sis, the popularityof the internal financ18 See esp. [8]. Gordon's views represent the most ing model will be found to reflect little explicit and sophisticated formulation of what might more than its ease of manipulationcom- be called the "bird-in-the-hand" fallacy. For other, elaborate, statements of essentially the same bined with the failure to push the analy- less position see, among others, Graham and Dodd [11, sis far enough to disclosehow special and p. 433] and Clendenin and Van Cleave [3]. how treacherousa case it really is. 19 We use the notation Ap(t) to avoid any confusion In particular, concentration on this between Gordon's purely subjective discount rate special case appearsto be largely respon- and the objective, market-given yields p(t) in Sec. I above. To attempt to derive valuation formulas sible for the widely held view that, even under uncertainty from these purely subjective disunderperfect capital markets,there is an count factors involves, of course, an error essentially This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:38:51 UTC All use subject to JSTOR Terms and Conditions 425 THE VALUATIONOF SHARES Hence, when we use a single uniformdiscount rate p as in (22) or (23), this rate shouldbe thought of as really an average of the "true"rates p(t) each weighted by the size of the expected dividend payment at time t. If the dividend stream is growing exponentially then such a weighted average p would, of course, be higher the greater the rate of growth of dividendsg since the greaterwill then be the portion of the dividend stream arising in the distant as opposedto the near future. But if all financingis assumed to be internal, then g = krp* so that given p*, the weighted average discount factor p will be an increasing function of the rate of retention kr which would run counter to our conclusionthat dividend policy has no effect on the currentvalue of the firm or its cost of capital. For all its ingenuity, however, and its seeming foundation in uncertainty, the argument clearly suffers fundamentally from the typical confoundingof dividend policy with investment policy that so frequently accompanies use of the internal financingmodel. Had Gordonnot confinedhis attention to this special case (or its equivalent variants), he would have seen that while a change in dividend policy will necessarily affect the size of the expected dividendpayment on the sharein any futureperiod,it need not, in the general case, affect either the size, of the total return that the investor expects duringthat period or the degree of uncertainty attaching to that total return. As should be abundantly clear by now, a change in dividend policy, given investment policy, implies a change only in the distributionof the total return in any period as between dividends and capital gains. If investors behave ration- ally, such a change cannot affect market valuations. Indeed, if they valued shares according to the Gordon approach and thus paid a premium for higher payout ratios, then holders of the low payout shareswouldactually realizeconsistently higher returns on their investment over any stated interval of time.20 Corporateearningsand investorreturns. -Knowing the relation of g to kp* we can answera question of considerableinterest to economic theorists, namely: What is the precise relation between the earningsof the corporationin any period X(t) and the total return to the owners of the stock during that period?2'If we let Gt(t) be the capital gains to the owners during t, we know that Dt (t) +Gt (t) = X(t) X(1 - kr)+U V( 26 ) 20 This is not to deny that growthstocks (in our sense)maywellbe "riskier"thannon-growthstocks. But to the extent that this is true, it will be due to the possibly greater uncertaintyattaching to the size anddurationof futuregrowthopportunitiesand henceto the size of the futurestreamof total returns quite apart from any questionsof dividendpolicy. 21 Note also that the aboveanalysisenablesus to deal very easily with the familiarissue of whethera firm'scost of equity capitalis measuredby its earnings/priceratioor by its dividend/priceratio.Clearly, the answeris that it is measuredby neither,except undervery specialcircumstances.For from(23) we have for the earnings/priceratio X(O) _p-kp* V (O) 1-k which is equal to the cost of capital p, only if the firm has no growth potential (i.e., p* = p). And from (24) we have for the dividend/priceratio D(O) V (O) g whichis equalto p only wheng = 0; i.e., from(25), either when k = 0; or, if k > 0, when p*